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October  2014, 10(4): 1041-1058. doi: 10.3934/jimo.2014.10.1041

On a risk model with randomized dividend-decision times

Zhimin Zhang 1,

1. 

College of Mathematics and Statistics, Chongqing University, Chongqing, 401331, China

Received  October 2012 Revised  December 2013 Published  February 2014

In this paper, we consider a perturbed compound Poisson risk model with a randomized dividend strategy. Assume that decisions on paying off dividends are made at some random observation times. Whenever the observed value of the surplus process exceeds a given barrier $b$, the excess value will be paid off as dividends. We assume that the Laplace transform of the individual claim size belongs to the rational family. When the time intervals between successive decision times follow exponential distribution, we present explicit expressions for the Gerber-Shiu function. We also extend the exponential assumption to Erlang and discuss the solution procedure.
Keywords: Erlang., integral equation, Gerber-Shiu function, Dividend-decision time, exponential.
Mathematics Subject Classification: Primary: 91B30; Secondary: 62P0.
Citation: Zhimin Zhang. On a risk model with randomized dividend-decision times. Journal of Industrial & Management Optimization, 2014, 10 (4) : 1041-1058. doi: 10.3934/jimo.2014.10.1041
References:
[1]

H. Albrecher, E. C. K. Cheung and S. Thonhauser, Randomized observation periods for the compound Poisson risk model: Dividends, Astin Bulletin, 41 (2011), 645-672.  Google Scholar

[2]

H. Albrecher, E. C. K. Cheung and S. Thonhauser, Randomized observation periods for the compound Poisson risk model: The discounted penalty function, Scandinavian Actuarial Journal, 2013 (2013), 424-452. doi: 10.1080/03461238.2011.624686.  Google Scholar

[3]

B. Avanzi, E. C. K. Cheung, B. Wong and J.-K. Woo, On a periodic dividend barrier strategy in the dual model with continuous monitoring of solvency, Insurance: Mathematics and Economics, 52 (2013), 98-113. doi: 10.1016/j.insmatheco.2012.10.008.  Google Scholar

[4]

B. De Finetti, Su un impostazione alternativa della teoria collectiva del rischio, Transactions of the XVth International Congress of Actuaries, 2 (1957), 433-443. Google Scholar

[5]

H. U. Gerber and E. S. W. Shiu, On the time value of ruin, North American Actuarial Journal, 2 (1998), 48-78. doi: 10.1080/10920277.1998.10595671.  Google Scholar

[6]

H. U. Gerber and E. S. W. Shiu, Optimal dividends: Analysis with Brownian motion, North American Actuarial Journal, 8 (2004), 1-20. doi: 10.1080/10920277.2004.10596125.  Google Scholar

[7]

A. E. Kyprianou, Introductory Lectures on Fluctuations of Lévy Processes with Applications, Springer-Verlag, Berlin, Heidelberg, 2006.  Google Scholar

[8]

S. Li, The distribution of the dividend payments in the compound poisson risk model perturbed by diffusion, Scandinavian Actuarial Journal, 2006 (2006), 73-85. doi: 10.1080/03461230600589237.  Google Scholar

[9]

S. Li and J. Garrido, On ruin for the Erlang(n) risk model, Insurance: Mathematics and Economics, 34 (2004), 391-408. doi: 10.1016/j.insmatheco.2004.01.002.  Google Scholar

[10]

S. Li and J. Garrido, On a class of renewal risk model with a constant dividend barrier, Insurance: Mathematics and Economics, 35 (2004), 691-701. doi: 10.1016/j.insmatheco.2004.08.004.  Google Scholar

[11]

X. S. Lin, G. E. Willmot and S. Drekic, The classical risk model with a constant dividend barrier: Analysis of the Gerber-Shiu discounted penalty function, Insurance: Mathematics and Economics, 33 (2003), 551-566. doi: 10.1016/j.insmatheco.2003.08.004.  Google Scholar

[12]

X. S. Lin and K. P. Pavlova, The compound Poisson risk model with a threshold dividend strategy, Insurance: Mathematics and Economics, 38 (2006), 57-80. doi: 10.1016/j.insmatheco.2005.08.001.  Google Scholar

[13]

X. S. Lin and K. P. Sendova, The compound Poisson risk model with multiple thresholds, Insurance: Mathematics and Economics, 42 (2008), 617-627. doi: 10.1016/j.insmatheco.2007.06.008.  Google Scholar

[14]

C. C. L. Tsai and G. E. Willmot, A generalized defective renewal equation for the surplus process perturbed by diffusion, Insurance: Mathematics and Economics, 30 (2002), 51-66. doi: 10.1016/S0167-6687(01)00096-8.  Google Scholar

[15]

Z. Zhang and X. Wu, Dividend payments in the Brownian risk model with randomized decision times, Acta Mathematicae Applicatae Sinica-English Series, (2013), accepted. Google Scholar

show all references

References:
[1]

H. Albrecher, E. C. K. Cheung and S. Thonhauser, Randomized observation periods for the compound Poisson risk model: Dividends, Astin Bulletin, 41 (2011), 645-672.  Google Scholar

[2]

H. Albrecher, E. C. K. Cheung and S. Thonhauser, Randomized observation periods for the compound Poisson risk model: The discounted penalty function, Scandinavian Actuarial Journal, 2013 (2013), 424-452. doi: 10.1080/03461238.2011.624686.  Google Scholar

[3]

B. Avanzi, E. C. K. Cheung, B. Wong and J.-K. Woo, On a periodic dividend barrier strategy in the dual model with continuous monitoring of solvency, Insurance: Mathematics and Economics, 52 (2013), 98-113. doi: 10.1016/j.insmatheco.2012.10.008.  Google Scholar

[4]

B. De Finetti, Su un impostazione alternativa della teoria collectiva del rischio, Transactions of the XVth International Congress of Actuaries, 2 (1957), 433-443. Google Scholar

[5]

H. U. Gerber and E. S. W. Shiu, On the time value of ruin, North American Actuarial Journal, 2 (1998), 48-78. doi: 10.1080/10920277.1998.10595671.  Google Scholar

[6]

H. U. Gerber and E. S. W. Shiu, Optimal dividends: Analysis with Brownian motion, North American Actuarial Journal, 8 (2004), 1-20. doi: 10.1080/10920277.2004.10596125.  Google Scholar

[7]

A. E. Kyprianou, Introductory Lectures on Fluctuations of Lévy Processes with Applications, Springer-Verlag, Berlin, Heidelberg, 2006.  Google Scholar

[8]

S. Li, The distribution of the dividend payments in the compound poisson risk model perturbed by diffusion, Scandinavian Actuarial Journal, 2006 (2006), 73-85. doi: 10.1080/03461230600589237.  Google Scholar

[9]

S. Li and J. Garrido, On ruin for the Erlang(n) risk model, Insurance: Mathematics and Economics, 34 (2004), 391-408. doi: 10.1016/j.insmatheco.2004.01.002.  Google Scholar

[10]

S. Li and J. Garrido, On a class of renewal risk model with a constant dividend barrier, Insurance: Mathematics and Economics, 35 (2004), 691-701. doi: 10.1016/j.insmatheco.2004.08.004.  Google Scholar

[11]

X. S. Lin, G. E. Willmot and S. Drekic, The classical risk model with a constant dividend barrier: Analysis of the Gerber-Shiu discounted penalty function, Insurance: Mathematics and Economics, 33 (2003), 551-566. doi: 10.1016/j.insmatheco.2003.08.004.  Google Scholar

[12]

X. S. Lin and K. P. Pavlova, The compound Poisson risk model with a threshold dividend strategy, Insurance: Mathematics and Economics, 38 (2006), 57-80. doi: 10.1016/j.insmatheco.2005.08.001.  Google Scholar

[13]

X. S. Lin and K. P. Sendova, The compound Poisson risk model with multiple thresholds, Insurance: Mathematics and Economics, 42 (2008), 617-627. doi: 10.1016/j.insmatheco.2007.06.008.  Google Scholar

[14]

C. C. L. Tsai and G. E. Willmot, A generalized defective renewal equation for the surplus process perturbed by diffusion, Insurance: Mathematics and Economics, 30 (2002), 51-66. doi: 10.1016/S0167-6687(01)00096-8.  Google Scholar

[15]

Z. Zhang and X. Wu, Dividend payments in the Brownian risk model with randomized decision times, Acta Mathematicae Applicatae Sinica-English Series, (2013), accepted. Google Scholar

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